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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators
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by Steve Hofmann, Carlos Kenig, Svitlana Mayboroda and Jill Pipher PDF
J. Amer. Math. Soc. 28 (2015), 483-529 Request permission

Abstract:

We consider divergence form elliptic operators $L= {-}\mathrm {div} A(x) \nabla$, defined in the half space $\mathbb {R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A(x)$ is bounded, measurable, uniformly elliptic, $t$-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation $Lu=0$, and we then combine these estimates with the method of “$\epsilon$-approximability” to show that $L$-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class $A_\infty$ with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in $L^p$, for some $p<\infty$). Previously, these results had been known only in the case $n=1$.
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Additional Information
  • Steve Hofmann
  • Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
  • MR Author ID: 251819
  • ORCID: 0000-0003-1110-6970
  • Email: hofmanns@missouri.edu
  • Carlos Kenig
  • Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois, 60637
  • MR Author ID: 100230
  • Email: cek@math.chicago.edu
  • Svitlana Mayboroda
  • Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
  • MR Author ID: 739839
  • Email: svitlana@math.umn.edu
  • Jill Pipher
  • Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
  • MR Author ID: 237541
  • Email: jpipher@math.brown.edu
  • Received by editor(s): February 10, 2012
  • Received by editor(s) in revised form: February 11, 2014
  • Published electronically: May 21, 2014
  • Additional Notes: Each of the authors was supported by the NSF
    This work has been possible thanks to the support and hospitality of the University of Chicago, the University of Minnesota, the University of Missouri, Brown University, and the BIRS Centre in Banff (Canada). The authors would like to express their gratitude to these institutions.
  • © Copyright 2014 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 28 (2015), 483-529
  • MSC (2010): Primary 42B99, 42B25, 35J25, 42B20
  • DOI: https://doi.org/10.1090/S0894-0347-2014-00805-5
  • MathSciNet review: 3300700